+1557 Let a and b be the roots of the quadratic x^2 - 5x + 3 = 2x^2 + 15x - 10. Find the quadratic whose roots are a^2 + a + 1 and b^2 + b + 1.
+9673 Simplifying, the quadratic equation is \(x^2 + 20x - 13 = 0\). Therefore, a + b = -20 and ab = -13.
Now,
\(\quad (a^2 + a + 1) + (b^2 + b + 1)\\ = (a^2 + b^2) + (a + b) + 2\\ = (a + b)^2 - 2ab + (a + b) + 2\\ = 400 + 26 - 20 + 2\\ = 408\)
Also,
\(\quad (a^2 + a + 1)(b^2 + b + 1)\\ = \dfrac{(a^3 -1)(b^3 - 1)}{(a - 1)(b - 1)}\\ = \dfrac{(ab)^3 - (a^3 + b^3) + 1}{ab - (a + b) + 1}\\ = \dfrac{(ab)^3 - (a + b)^3 + 3ab(a+b) + 1}{ab - (a + b) + 1}\\ = 823\)
Hence, the required equation is \(x^2 - 408x + 823 = 0\).
